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Asymptotic of Enumerative Invariants in \({\mathbb {C}}P^2\)

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Abstract

In this paper, we give the asymptotic expansion of \(n_{0,d}\) and \(n_{1,d}\), where \((3d-1+g)!\, n_{g,d}\) counts the number of genus g curves in \({\mathbb {C}}P^2\) through \(3d-1+g\) points in general position and can be identified with certain Gromov–Witten invariants.

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References

  1. Di Francesco, P., Itzykson, C.: Quantum intersection rings. arXiv:hep-th/9412175

  2. Fukaya, K., Ono, K.: Arnold conjecture and Gromov–Witten invariant. Topology 38(5), 933–1048 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Li, J., Tian, G.: Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, Topics in symplectic 4-manifolds (Irvine, CA, 1996), 47–83, Int. Press Lect. Ser., I, Int. Press, Cambridge, MA (1987)

  4. Mcduff, D., Salamon, D.: J-holomorphic curves and quantum cohomology. University Lecture Series, vol. 6, pp. viii+207. American Mathematical Society, Providence, RI (1994)

    MATH  Google Scholar 

  5. Pandharipande, R.: A geometric construction of Getzler’s elliptic relation. Math. Ann. 313(4), 715–729 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. J. Differ. Geom. 42(2), 259–367 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. Math. Res. Lett. 1(2), 269–278 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ruan, Y., Tian, G.: Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130(3), 455–516 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zinger, A.: On asymptotic behavior of GW-invariants, preprint (2011)

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Acknowledgements

We thank the referee for numerous comments which are helpful in improving the exposition of this paper.

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Authors and Affiliations

  1. Beijing International Center for Mathematical Research and School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Gang Tian & Dongyi Wei

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  1. Gang Tian
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  2. Dongyi Wei
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Correspondence to Dongyi Wei.

Additional information

Gang Tian: Supported partially by Grants from NSF and NSFC.

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Tian, G., Wei, D. Asymptotic of Enumerative Invariants in \({\mathbb {C}}P^2\). Peking Math J 1, 125–140 (2018). https://doi.org/10.1007/s42543-018-0004-4

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  • DOI: https://doi.org/10.1007/s42543-018-0004-4

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